OBLIQUE TBIANGLES.] MA THEM ATICS. SPHERIC AL TRIGONOMETRY. 



661 



present article, are expressed as products, and con 

 sequently are adapted for logarithmic calculations. Th< 

 cases, as we have already seen, are the following : 



(1) Given a and 6, find c, A and B 



Cos. c is given by (19), then Cos. A is given by 

 (23), and cos. B by (24) 



(2) Given c and a, find b, A and B 



Cos. 6 is given by (19), then cos. A is given by (23), 

 and cos. B by (24) 



(3) Given a and b, find b, c, and A 



Cos. A is given by (21), then cos. 6 is given by (22), 

 and tan. c by (23) 



(4) Given a and A, find b, c, and B 



Sin. B is given by (21), then cos. b is given by (22), 

 and cos. c by (19) 



(5) Given c and A, find a, b, and B 



Sin. b is given by (17), then cos. a is given by (19), 

 and cos. B by (22) 



(6) Given A and B, find a, b, and c 



Cos. c is given by (20), cos. a by (21), and cos. * 

 by (22). 



It will be observed that each of the above determi- 

 nations is clearly unambiguous, except the determination 

 of B in (4), and 6 in (5), for these are the only two deter- 

 minations made by means of sines ; for which reason, if 

 B' and b' are the values less than 90 .which satisfy (4) 

 and (5), then 180 B', and 180 b' also satisfy (4) 

 and (5) ; and hence it would seem that in the former 

 case there would in general be two values of b and two 

 of f, corresponding to B' and 180 B' respectively ; 

 and in the latter case, that there would be two values of 

 a given by (3), and therefore two values of B given by 

 (6). If more closely considered, however, it will appear 

 that there is really no ambiguity in case (o). We will 

 consider the cases separately. 



In case (4) we have given A and o. Now AB and AC 

 (Fig. 4), being produced, meet at A', where ABA' and 

 A CA'are eaoh of arc jj g 4. 



IflO" (see R[hen'cal ,_ 



Geometry, pp.008, ri!*!) 

 and the angle at A' is 

 e, ual to the angle at A. 

 Hence the angle A and 



the side o belong equally to triangle ABC and to A'B C. 

 And if we take the value of B less than 90 to be A B C, 

 then C B A' is 180 B', the second value indicated by 

 the solution. 



In case (5) from (10) it appears that tan. a = sin. 6 

 tan. A. 



Now sin. 6 is always positive, and hence the sign 

 of tan. a must be the same as that of tan A. Hence if 



Fig. s. 



A 790 a must be 7 90, 

 and if o 7 90 A must be 

 ^ 90, and A is given, 

 hence only one of the two 

 values of a is admissible. 

 This also follows from 

 geometrical considera- 

 tions. Let C be the right 

 angle, then C A and C B, 

 when produced, meet in C'; then since we have given A 15 

 (<) and B A C (A) we determine b,i.t., cA from the equa- 

 tion or 180 6, i.e., A C"; but A C 7 belongs to a triangle 

 on which the angle B A CX is not A, but 180 A, and 

 .'. the value AC* is inadmissible. 



THE SOLUTION OF OBLIQUE-ANGLED TRIANGLES. 

 (9). To enumerate the Caset of Oblique-Angled Triangles. 



There are six cases of oblique-angled triangles, viz. : 



(1). Given three sides, e. g., a. b. c 



(2). Given two sides and the included angle, e. g., a.b. 

 and C. 



(3). Given two sides, an angle opposite to one of them, 

 . 6. A 



(4). Given one side and the two adjacent angles, e.g., 

 A. B. c 



(5). Givon one Hide, the opposite angle and another 

 , e.g., A. C. c 



(6). Given the three angles, e. g., A. B. C 

 As in the case of right-angled triangles, these six cases 

 are analogous to the four cases of plane oblique-angled 

 triangles (p. 641). But the fourth case of a planetri- 

 angle diverges on to the fourth and fifth of the spherical 

 triangle, owing to the circumstance that A + B + C is 

 not known in the case of the spherical triangle, whereas 

 in the plane triangle A -f B + C = 180. For the same 

 reason case (6) is peculiar to the spherical triangle. 



(10). To Solve the First Case of Obliqiie- Angled Triangles. 

 We can obtain A from either of the formulas (5), (6), 

 or (7), and then can obtain B and C from similar for- 

 mulas. Of these formulas (7), which gives tan. -7,-, is the 



most convenient if we wish to find both of the other 

 angles. Compare the analogous case of Plane Triangles. 

 p. 641. 



(11). To Solve the Second Case of Oblique-Angkd 



Trianyles. 



In this case we will suppose that we have given a b and 

 C. Then from formulas (13) and (14) we can determine 

 A+B ,A-B 

 2 and ^ and hence A and B ; and then, 



knowing A and B, we can determine c from formula (1). 

 If, however, we wish to determine c directly, i. e , 

 independently of A and B, we can effect our object 

 by introducing a subsidiary angle in a manner analogous 

 to the corresponding case of plane triangles. (See pp. 

 623 and 642). Thus, from formula (3) we have 



Cos C _ cos - c CO8 - a cog - & 



sin. a sm. 6 



Cos. c "* cos. a cos. 6 + sin. a sin. 6 cos. C 

 f ? . 1 COB. c = cos. a cos. 6 sin. o sin. 6 cos. C 

 = 1 cos. a cos. b + sin. o sin. 6 



sin. a sin. 6 (1 -j- cos. C) 



= 1 (COB. a cos. 6 sin. a sin. b) 



sin. a sin. 6 (1 -f- cos. C) 



= 1 cos. (a + 6) - sin. a sin. 6 

 (1 + cos. C) 



Now 1 cos. A = 2 sin. 1 -=- and 1 + cos. A = 2 cos. 1 -- 



.c ,0+6 , C 



sin. ! A = Bin.' sin. a sin. o cos. 2 - 



mm 2 



Assume sin. ! = sin. a sin. 6 cos."^- 



sin. 



.'. 8i n.'|-sin.^_ 8i n.* 



{** 



= 2 sin. - ( - 



x.*j(sji 



mt 



im.tfWsm.fL+J'+sin. \ 



I cos. 



-+ 



)-K : 



\ 1 / , 



'+& 



!.'-= sin. 



sin. _-'_. 



Thia latter method is very much easier than the 

 brmer ; for by this we only require five logarithms, 

 whereas by that we require eight, for the determination 

 .f c. 



12). To Solve the Third Case of Oblique- Angled Triangles. 



In this case we will suppose that we have given a, b, A 



Then we obtain sin. B by formula (1) ; and knowing a, l>, 



